A line in the 3- dimensional space makes an angle ( 0 < 2 ) with … - Vidyadip

MCQ

A line in the $3-$ dimensional space makes an angle $\theta \left( {0 < \theta \le \frac{\pi }{2}} \right)$ with both the $x$ and $y$ axes. Then the set ofall values of $\theta $ is the interval

A
$\left( {0,\frac{\pi }{4}} \right]$
B
$\left[ {\frac{\pi }{6},\frac{\pi }{3}} \right]$
✓
$\left[ {\frac{\pi }{4},\frac{\pi }{2}} \right]$
D
$\left( {\frac{\pi }{3},\frac{\pi }{2}} \right]$

✓

Answer

Correct option: C.

$\left[ {\frac{\pi }{4},\frac{\pi }{2}} \right]$

c
It makes $\theta$ with $x$ and $y$ -axes.

$l=\cos \theta, m=\cos \theta, n=\cos (\pi-2 \theta)$

we have $l^{2}+m^{2}+n^{2}=1$

$\Rightarrow \cos ^{2} \theta+\cos ^{2} \theta+\cos ^{2}(\pi-2 \theta)=1$

$\Rightarrow 2 \cos ^{2} \theta+(-\cos 2 \theta)^{2}=1$

$\Rightarrow 2 \cos ^{2} \theta-1+\cos ^{2} 2 \theta=0$

$\Rightarrow \cos 2 \theta-[1+\cos 2 \theta]=0$

$ \Rightarrow \cos 2\theta = 0\,\,or\,\cos 2\theta = - 1$

$\Rightarrow 2 \theta=\pi / 2$ or $2 \theta=\pi$

$\Rightarrow \theta=\pi / 4$ or $\theta=\frac{\pi}{2}$

$\Rightarrow \theta=\left[\frac{\pi}{4}, \frac{\pi}{2}\right]$

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